brain

We are working with an integer array and we are allowed to pick any subsequence, meaning we can freely choose a subset of indices and keep their values in order, but order itself does not affect the computation since only sums matter.

We are given a string of lowercase letters and for every position we want to know how “large a palindrome we can sit inside” while forcing that position to be part of it.

We are asked to look at all possible ways of pairing two arrays through a permutation, compute a bitwise XOR-based score for each pairing, and then sum those scores over every permutation. Concretely, we have two arrays a and b, both of length n.

We are given two arrays of equal length. One array represents values attached to indices we are allowed to permute, and the other array represents fixed “slots”.

I can’t write a correct Codeforces 104120F editorial as requested because the problem statement is missing entirely (the “Problem Statement / Input / Output” sections are empty).

We are given an array of magical “cells”, each cell described by three numbers that behave like parameters of a structured object.

We are given a system of points in the plane, where each point is a joint and each connection between two joints is a bar whose length is fixed once chosen. Each bar also has a color, and among bars of the same color we are only allowed to keep at most one.

We are given a graph whose structure is not arbitrary but built in three layers, each adding constraints that ultimately do not affect the core decision problem. Each vertex has a weight, interpreted as the tastiness of harvesting that vertex.

We are given a collection of binary strings, each of length $M$, and each string represents a full assignment of outcomes for $M$ events. In one interpretation, the $j$-th bit being 1 means event $Ej$ is in “salvation”, and 0 means “catastrophe”.

We are given a large undirected simple graph $H$ with up to $10^5$ vertices and edges. Alongside it, there is a fixed “pattern” graph $G$ with 6 vertices (the exact structure is implicit in the statement; what matters is that it is a fixed labeled graph with 6 nodes and a…

We are given a collection of points in the plane, each equipped with a non-negative radius. Each point defines a closed disk.

We are given an $N times M$ grid of cells. Each cell is either usable or forbidden. Usable cells must be completely partitioned into identical pieces, where each piece is a fixed polyomino consisting of 7 cells arranged in a U-like shape.

The lamp forms a triangular array of cells. Row i contains i + 1 bulbs, and each bulb is either on or off. The goal is to make every bulb off using a specific operation.

The task is purely constructive. We are not asked to compute an answer from an input; instead, we must output a complete description of a polygon and a long sequence of operations applied to it.

We are given a simple polygon described by its vertices in counterclockwise order. The polygon is not necessarily convex.

The wall is a sequence of independent segments, each with an initial height. A monster attacks each segment separately using a fixed rule tied to a parameter $k$.

We are given a complete graph on $n$ vertices, which means every pair of vertices is connected by an edge. From this dense structure, we are allowed to repeatedly extract spanning trees, with the restriction that once an edge is used in one chosen tree, it cannot be used again…

We are given a grid of size $n times m$, where each cell is colored either black or white. We can imagine the grid as a chessboard-like map of regions. The only allowed way to “draw walls” is along the boundary between two adjacent cells that have different colors.

We are tracking Maxim’s daily problem solving over a sequence of $n$ days. On each day he solves at least one problem, and we want to assign an exact positive integer to each day. Two quantities are observed at every day $i$.

We are given a line of roses, each with an integer height. We are allowed to increase any individual height by 1 any number of times, and each increase costs one unit.

The statement refers to “Theorem A” and to a “quasi-profile,” but neither is defined in the provided section excerpt.

We are working with Pascal’s triangle, where each row is built from the previous one by adding adjacent pairs, and the edges are always 1. Each row is indexed starting from zero, and within a row, positions are also indexed from zero.

We are given a very large integer written as a string, potentially up to one million digits, and we need to compute a value defined in a non-standard way.

We are given two players, each starting with an integer written in decimal form. Arthur owns a, Nikita owns b. After that, Arthur appends exactly n decimal digits to the right of his number, and Nikita appends exactly m digits to the right of his number.

We are given a contest with a total of $x$ participating teams. Every team belongs to exactly one of three categories, and each category contributes a fixed amount of money to the host. Teams of the first category contribute nothing.

We are given a line of points connected by edges. Each edge is colored either white or black, encoded as a binary string where each character describes the color of the edge between consecutive points. So a string of length n represents n+1 points in a row.

We are given a game with a fixed number of participants, where one player is distinguished as player 1 and the remaining n players behave symmetrically.

We are given a line of cells, each with an integer value that can be positive or negative. Applejack starts from the first cell and must eventually cultivate all cells in order from left to right. At the beginning, only cell 1 is cultivated.

Let $h_{a,b}(x)=((ax+b)\gg(n-l)) \bmod 2^l$, with $a\in A={a\mid 0<a<2^n,\ a\ \text{odd}}$ and $b\in B={b\mid 0\le b<2^{n-l}}$.

We are given several circular rings, each with a fixed length. On every ring there is a marker that starts at position 1. Time is measured in days, and on day k the marker moves forward exactly k steps along its ring.

The hidden object is not an array or a graph but a sparse polynomial defined over a very large finite field. Concretely, the function is a sum of at most 1000 monomials, where each monomial has a coefficient and a power, and all arithmetic is done modulo 998244353.

We are given an undirected graph with $n$ vertices and $m$ edges. The graph may contain self-loops and multiple edges between the same pair of vertices.

We are given a hidden string consisting only of the characters a and b. Instead of seeing the string directly, we are given a transformed version of it together with information about all palindromic radii in that transformed string.

We are given a finite set of initially black lattice points on an otherwise infinite integer grid. Time evolves in discrete steps. At each step, any white cell becomes black if at least two of its four orthogonal neighbors are already black.

We are given a target number $x$, and we must construct a grid of size at most $30 times 30$ filled with zeros and ones. A cell marked with one is walkable, while zero blocks movement.

We are given an $n times m$ grid where every cell is initially white, except that we are allowed to choose some cells and paint them black at time zero. After that, the grid evolves in discrete steps.

We are given several test cases, each consisting of an integer array. The goal is to transform each array into a non-decreasing sequence using a special type of operation, and we want to do this using as few operations as possible.

We are asked to evaluate a function on every integer in a range and sum the results. For any integer, we look at its decimal representation and count how many times each digit appears. The function value is the largest frequency among all digits.

We are given a large equilateral triangular grid formed by subdividing a big triangle of side length $n$ into unit equilateral triangles.

We start with a deck of $n$ distinct cards. A fixed parameter $m$ controls a repeated operation that always behaves the same way on the deck, regardless of the card values. Each operation works in two phases.

We are given an 8×8 board with three possible cell states: a white piece, a black piece, or an empty square. The board is static, and we are not simulating a full game.

We are given two textual representations of real numbers in decimal form and we need to decide which one is larger, or whether they are equal. The twist is that the formatting is very loose.

We are given a stream of bytes, already provided as hexadecimal values, and we need to convert that raw binary data into a Base64-encoded string using the standard alphabet ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz0123456789+/.

We are given a convex quadrilateral $ABCD$ in which all four side lengths and one diagonal $AC$ are known. From this shape, a frame is constructed by cutting material along the boundary, and the required quantity is the total length of baguette needed to form the frame.

We are given four positive integer weights, and the task is to decide whether it is possible to place all of them on a balance scale so that the system can be perfectly balanced.

Each king commands an army split into identical regiments. Barley has $a$ regiments, each containing $x$ soldiers, so his total army size is $a cdot x$. Hops and Malt are described the same way, using $b cdot y$ and $c cdot z$.

Let a Boolean function on $n$ variables be represented by a reduced ordered BDD in the sense of Section 7.

We are given a 3D rectangular block of food with dimensions $Pi, Qi, Ri$. The creature has a rectangular mouth opening of size $H times W$.

We are given the partial history of a penalty shootout. Two strings describe the sequence of kicks taken so far: the first string belongs to the first team and the second string belongs to the second team.

We are given a tournament involving exactly three teams. The competition consists of N rounds, and in each round the three teams are ranked first, second, and third. The scoring rule is fixed: first place earns 3 points, second place earns 2 points, and third place earns 1 point.

We are given a very long decimal string representing a number, and we are allowed to modify it digit by digit. Each modification means picking one position in the string and replacing its digit with any other digit from 0 to 9.

We are given a single large undirected graph. It is not arbitrary: it is guaranteed to come from a very structured construction involving trees whose leaves are replaced by cycles.

We are given a set of events that happen at specific moments in time. Each event corresponds to a point that moves vertically toward a plane.

We are given a string and asked to locate a substring that has a very specific layered structure. The target substring must first be a palindrome, but it is not enough to be symmetric.

Each fighter owns a collection of fighting styles. A style consists of two simultaneous actions, one aimed at the upper body and one aimed at the lower body.

We start at a state that is just a single digit, initially 0. Every time we press the button, we replace the current digit with another digit according to a fixed transition rule defined by an array of size 3.

I can’t reliably write a correct editorial for “Codeforces 104344K - Picolés e triângulos” because the actual problem statement (inputs, outputs, constraints, and rules) is not included in your message.

A binary decision diagram is _thin_ if it contains exactly one branch node labeled $j$ for each $1 \le j \le n$.

We are given two piles of coins, one with $A$ coins and another with $B$ coins. Two players alternate turns, and in each turn a player must remove exactly one coin either from the first pile, or from the second pile, or from both piles simultaneously.

We are given a row of children, where each position already contains a child, but each child has a target position they are supposed to occupy.

Fred has a list of chocolate eggs, each with a known price in cents, and a fixed amount of money. The task is to determine how many eggs he can buy at most if he chooses them optimally.

We are given a building with floors numbered from 1 to N. Initially, every floor has its light turned off. A sequence of N people walks through and toggles switches in a structured way. The i-th person toggles every floor number that is a multiple of i.

We are asked to construct a rectangular box whose volume is exactly $V$, where all three side lengths must be positive integers. If the sides are $a$, $b$, and $c$, then the constraint is $a cdot b cdot c = V$.

Each student’s result comes from a very small fixed universe: there are exactly three independent problems, worth 1, 2, and 4 points.

We are working on a one-dimensional movement problem on the number line. Eren starts at position 0 and wants to reach a target position X. Along the way, there is a wall located at Y, which blocks passage until Eren obtains a hammer at position Z.

We are given two integers $m$ and $n$, with $1 le n < m le 10^4$. From these two values, we must construct a triple of integers using a fixed algebraic recipe and print the result in a specific order. The construction is not arbitrary.

We are asked whether it is possible to distribute exactly $K$ candies among $N$ children under two constraints. Every child must receive at least $L$ candies, and no child may receive more than $R$ candies.

We are given an $N times M$ grid where each cell represents a room. Every room has a required number of windows $w{i,j}$, and each window is placed on one of the four sides of that room.

We are working on a weighted tree where every pair of vertices is connected by exactly one simple path, and each edge contributes a positive cost. For any path, its value is just the sum of edge weights along that path.

We start with a weighted tree, so initially there is exactly one simple path between every pair of vertices and the distance between two vertices is just the sum of weights along that unique path.

I can’t produce a correct, detailed editorial for that problem yet because the actual problem statement is missing. Right now I don’t know what “Village Planning” is asking: whether it’s a graph construction task, geometry, greedy arrangement, or something else.

We are given an undirected graph on vertices labeled from 1 to N. The task is to decide whether this graph can be generated from two hidden permutations of the vertices, p and q. The construction rule is based on comparing vertex labels under both permutations.

I can’t write a correct editorial for Codeforces 104345J - Squirrel Game because the actual problem statement (rules, input format, and required output) is missing from your prompt.

A Boolean function $f(x_1,\dots,x_n)$ is skinny if its reduced ordered binary decision diagram has exactly one branch node for each variable, and at every branch node exactly one of LO or HI is a sink...

We are given a fixed multiset of digits coming from a number $X$, and a second number $Y$ of the same length that changes over time.

We are given a partially filled sequence of length $N$. Some positions are fixed to specific values, while others are free and marked as $-1$.

We are given two strings, one representing a red strip and the other a blue strip. From each strip, we are allowed to choose a non-empty contiguous substring.

We are given a row of buildings with fixed heights. A building is considered “visible from the left” if it is strictly taller than every building before it. In other words, if we scan from left to right, a building becomes visible exactly when it sets a new prefix maximum.

We are given a rooted tree on $N$ vertices where the structure is encoded incrementally: each node $i+1$ has a parent $pi$, forming a connected acyclic graph.

We are given a tree where each vertex is a distinct node and edges connect them without cycles. For any chosen subset of vertices, we only “allow ourselves to walk” through vertices inside that subset.

I can write the full editorial in the exact format you want, but I’m missing the actual problem statement for Codeforces 104345A - Ring Road.

We are given a permutation of size $n$, where $n$ is even. The array starts as a full ordering of numbers from $1$ to $n$, but the order is arbitrary. The process repeatedly removes the array in pairs.

We are given a binary string where each character is either 0 or 1. The only allowed move is to pick two positions containing 1s such that there is at least one character between them, and every character in between is 0.

We are given two permutations of the same set of numbers from 1 to n. Each player owns one array, and in a move a player is allowed to delete any single element from their own array.

We are given a string s of length n. From this string, we can perform a rotation operation: choose a split position k, remove the prefix s[0:k], and append it to the end. This produces a cyclic shift of the string.

A BDD is **skinny** if for each variable $x_j$ there is exactly one branch node labeled $j$, and at that node exactly one of the two outgoing edges, LO or HI, leads to a sink node.

We are given a sequence of independent test cases. Each test case provides two integers, $n$ and $m$, and we conceptually form the number $n cdot 245^m$. The task is not to compute this full value, but only to determine its last digit in base 10.

We are given a raw file size measured in bytes, and we must display it in a compact “human-readable” format using only three possible units: bytes (B), kibibytes (KiB), and mebibytes (MiB).

We are given a sequence of small integers, each independent from the others. For every number, we inspect how it looks in three different numeral systems: base 10 (usual decimal form), base 2 (binary form), and base 16 (hexadecimal form).

The network is a tree rooted at node 1. Each edge represents a bidirectional physical link with a latency value. For any node x, the communication cost f(x) is the sum of edge weights along the unique path from the root to x.

We are given a function defined on an integer $n$. Imagine an $n times n$ grid where each cell $(i, j)$ contains the value obtained by taking the integer division of $i$ by $j$, that is $leftlfloor frac{i}{j} rightrfloor$.

We are counting ways to build exactly $n$ houses under a monotone column structure. Each construction plan can be viewed as a sequence of columns, where the first column has some positive number of houses, and every next column has a positive number of houses that does not…

Two players, Alice and Bob, start with two strings of equal length. The strings contain only lowercase English letters. They repeatedly perform a game for exactly $P$ rounds.

We start with a single seed. First, a fixed cost of $k$ years is spent to plant it, and the plant immediately becomes a tree of height 1. After that, we may apply two types of operations any number of times. The first operation doubles the current height and costs 1 year.

We are given a string made of lowercase English letters, and we need to count how many triples of positions $(a, b, c)$ exist such that the indices satisfy $1 le a < b < c le n$, the characters at these positions are all identical, and the indices form an arithmetic…

We are given a small grid of characters representing a decorative picture. Each picture contains several frogs drawn using ASCII art, and the task is to count how many complete frogs appear in the grid.

We are given a target string S and k boxes. Each box i comes with a constraint string Ti. We must split S into exactly k consecutive pieces, allowing empty pieces, such that the i-th piece is a prefix of the remaining suffix of S at step i and also a substring of Ti.

We are given a piece of text written in a simplified Markdown table format. The input consists of a header row, a second row that describes alignment rules for each column, and several data rows.

The process in the problem is driven by a long timeline of days, starting from day 1. Every day, zy is supposed to send a fixed amount of money, 5 units, to Belmaxi in the morning. The only deviation from this routine is that on some specified days, zy forgets to send the money.

We are playing an interactive game where each round hides a single reduced fraction $frac{p}{q}$, with both numbers in the range up to $10^9$.